\chapter{Introduction}
\label{chap:intro}

Millions of stars over the course of the Galaxy's history contributed
material to the cloud of gas and dust that became the Solar System (SS).
The cumulative ejecta from these stars mixed together to form our Sun,
the planets, and us, and this mixing was quite efficient, both elementally
and (at least) isotopically (See Appendix \ref{appendix:chondrites} for more
information about mixing).
When the materials experience various physical and chemical processes during
the formation of Solar System there are certain fractionated effects, again
both elementally and isotopically (See Appendix \ref{appendix:fractionation}
for more information about fractionation.)
That means the compositions of different objects in Solar System are different.
Studying these elemental and isotopic anomalies of various samples would 
give information about early
Solar System and then help us to understand the its formation.

%----------------------------------------------------------------------------
\section{FUN CAIs}\label{sect:fun}
%----------------------------------------------------------------------------

Calcium-aluminum-rich Inclusions (CAIs) are sub-millimeter to centimeter-sized
refractory objects found in carbonaceous chondrite meteorites (see Fig. 
\ref{fig:cais}).

\begin{figure}[ht!]
\centering
\includegraphics[width=\figuresize\textwidth]{figures/cais}
\caption{
Calcium-aluminum rich inclusions (CAIs) are found in carbonaceous chondrites, 
like the Allende meteorite pictured here (53 mm-long sample). The CAIs are 
the oldest materials to have formed in the Solar System, beating the 
Earth by 50 to 100 million years.
Caption from Jeffrey Taylor.
}
\label{fig:cais}
\end{figure}

Lead-lead dating (see Appendix \ref{appendix:chronology} for a brief 
description of chronology)
shows that they are among the oldest solid condensates in the early solar
system, if in fact not the oldest 
(e.g., \cite{2002Sci...297.1678A,2008ApJ...675L.121C}).
This is mainly because they are refractory and condensed relatively early
during the solar system formation at a high temperature 
(e.g. Al with a condensation temperature of 1650K). 
CAIs contain evidence of now extinct short-lived radioisotopes (e.g. $\al{26}$,
$\ca{41}$, and $^{182}$Hf) synthesized in one or multiple stars and then added
to the protosolar molecular cloud before or during its collapse.
Take $\al{26}$ ($t_{1/2} ~ 0.7$ Ma) as an example. 
The majority of CAIs contain high abundance
of radiogenic $\mg{26}$, the decay product of $\al{26}$, corresponding to an
inferred initial $\al{26}/\al{27}$ ratio of ~$(4.5-5.5) \times 10^{-5}$ (see
Fig. \ref{fig:al_mg} in Appendix \ref{appendix:chronology}).
This means these CAIs must have formed within a few million years after 
the fresh $\al{26}$ was produced and injected into proto solar system. 
Fig. \ref{fig:connelly} shows the demonstration of the formation timeline
for different objects.

\begin{figure}[ht!]
\centering
\includegraphics[width=\figuresize\textwidth]{figures/connelly}
\caption{
Timeline for chondrites and achondrites. Figure from Connelly et al. 2001.
}
\label{fig:connelly}
\end{figure}

However there a rare subset of refractory grains (e.g. hibonite) and 
inclusions (FUN CAIs) that have
low initial $\al{26}/\al{27}$ ratios ($<5\times 10^{-6}$) 
[Holst et al. 2103].
FUN CAIs are a small subset of CAIs with Fractionated and Unknown Nuclear
effects. Hibonite ((Ca,Ce)Al$_{12}$O$_{19}$) is a common mineral in the CAIs
found in some chondritic meteorites. Besides the feature of low $\al{26}$, 
both FUN CAIs and Hibonite grains carry isotopic anomalies in neutron-rich 
isotopes (see Fig.  \ref{fig:correlation}). 
In the meanwhile, both FUN CAIs and regular CAIs show $\oxygen{16}$-rich
in the three-isotope oxygen diagram (Fig. \ref{fig:oxygen_fun}).
So the puzzling question is: when did these objects form?
%
\begin{figure}[ht!]
\centering
\includegraphics[width=\figuresize\textwidth]{figures/oxygen_fun}
\caption{
Three-isotope oxygen diagram of oxygen–isotope compositions of individual 
minerals in the Allende STP-1 FUN CAI. Similarly to the majority of 
FUN CAIs, the oxygen–isotope compositions of anorthite, spinel, 
hibonite, and most Al,Ti-diopside grains in STP-1 plot along a 
mass-dependent fractionation line defining an initial $\Delta\oxygen{17}$ value 
of ∼ −24‰, that is, similar to the oxygen–isotope composition of 
canonical CAIs and that of the Sun. Oxygen–isotope compositions of 
melilite and some of the Al,Ti-diopside grains plot along a line with 
a slope of ∼1, suggesting subsequent isotope exchange with a 
$\oxygen{16}$-depleted 
gaseous reservoir. In contrast with most FUN CAIs from CV chondrites 
characterized by $\oxygen{16}$-poor compositions of melilite and anorthite, 
melilite in STP-1 shows a range of $\Delta\oxygen{17}$ values, 
whereas anorthite is 
uniformly $\oxygen{16}$-rich. These observations indicate that STP-1 is more 
pristine than all previously known FUN CAIs. Figure and caption from
[Holst et al. 2013].
}
\label{fig:oxygen_fun}
\end{figure}

From $\al{26}$ point of view, it may be either before regular CAIs formed
when few fresh $\al{26}$ has been injected into the proto solar system, or
after regular CAIs formed when a large portion of $\al{26}$ has decayed. 
Both possibilities cannot be confirmed or ruled out so far.

So studying the correlation of neutron-rich iron-group isotopes such as
$\ca{48}$ and $\ti{50}$ in FUN CAIs and Hibonite is another and important
way to explore the early history of the solar system.
The notation in Fig. \ref{fig:correlation} is defined as:

\begin{equation}
\delta \ca{48} / \ca{40} \equiv \left[ \ 
  \frac{ (\ca{48} / \ca{40})_{sample} }{ (\ca{48} / \ca{40})_{standard} } - 1
  \ \right] \times 1000 
\label{eq:delta}
\end{equation}

\begin{figure}[ht!]
\centering
\includegraphics[width=\figuresize\textwidth]{figures/correlation}
\caption{
Isotopic anomalies in the neutron-rich isotopes $\ca{48}$ and $\ti{50}$ 
measured in FUN and hibonite-bearing inclusions. The isotopic ratios are
expressed as $\delta$ values: deviations from the normal ratios are in
permil. Isotopic anomalies are qualitatively correlated in that most
inclusions with $\ca{48}$ excesses have $\ti{50}$ excesses and vice versa.
Figure and caption from \protect\cite{MeyerZinner}.
}
\label{fig:correlation}
\end{figure}

Studying the isotopic correlations will help constrain the Solar 
System formation in the nucleosynthetic aspect.

%----------------------------------------------------------------------------
\section{Neutron-Rich Iron-Group Isotopes}\label{sect:nrich}
%----------------------------------------------------------------------------

The most neutron-rich stable iron-group isotopes (namely, $^{48}$Ca,
$^{50}$Ti, $^{54}$Cr, $^{58}$Fe, and $^{62,64}$Ni) present an important
challenge for nucleosynthesis theory, Galactic chemical evolution studies,
and for our understanding of the birth of the Solar System.
With the exception of $^{48}$Ca, they are significantly produced by
slow neutron capture in massive stars (the weak {\em s}-process)
(e.g., \cite{2007ApJ...655.1058T}).  Such processing occurs primarily in
the large helium-burning convective core that follows core hydrogen burning.

The short beta-decay half of $^{45}$Ca
(162.2 days) and even shorter half of $^{47}$Ca (4.536 days)
prevent $^{48}$Ca from being produced in the same environment.  The
site of $^{48}$Ca synthesis is suspected to be a subset of
thermonuclear supernovae (type Ia supernovae)
that achieve high enough densities to have significant electron capture
during the explosion (e.g., \cite{1996ApJ...462..825M,1997ApJ...476..801W}).
Because the matter in these explosions has low entropy per nucleon
(typically less than 0.1 $k_B$, where $k_B$ is Boltzmann's constant,
per nucleon), the nuclear abundances are characterized by an overabundance
of heavy nuclei relative to nuclear statistical equilibrium.  This permits
production of abundant $^{48}$Ca when nuclear statistical equilibrium would
require even larger production of $^{66}$Ni, an isotope with nearly
the same degree of neutron richness \cite{1985ApJ...297..837H}.  

The type Ia supernovae that make $^{48}$Ca also make $^{50}$Ti, $^{54}$Cr,
and the other neutron-rich iron-group isotopes.  The yields depend in large
measure on the degree of neutron-richness achieved during the explosion.
This, in turn, depends on the weak interaction rates on the nuclei.  The
interesting challenge for nucleosynthesis theory, then, is to understand
the complex dynamics of the nuclear abundances as they shift due to both
weak interactions and changing temperature and density.

Because the neutron-rich iron-group isotopes are made in at least two
different sites, they also present a challenge to Galactic chemical
evolution studies.  Massive stars live and die quickly on astronomical
timescales.  For example, a star 25 times as massive as the Sun may be
born, live, and die in a core-collapse supernova event over a time span
of only ~7 million years (e.g., \cite{2007ApJ...655.1058T}).
A type Ia supernova is thought to be
the explosion of a white dwarf star that has accreted enough mass to
trigger a thermonuclear runaway that disrupts the entire star.  The white
dwarf formed from a lower mass star (less than roughly eight times the
mass of the Sun) and then had to accrete matter from a companion star.
The timescale for this to happen is hundreds of millions of years.  The build
up of abundance of an isotope like $^{50}$Ti in the Galaxy is coming from
two different sources operating on two different timescales while the build
up of the $^{48}$Ca abundance is apparently happening only on the longer
timescale.  The interesting challenge for Galactic chemical evolution
is, then, to follow the relative build up of these isotopes over time,
especially $^{48}$Ca compared to its sister neutron-rich iron-group species.

These differing formation scenarios and different production timescales
for the neutron-rich isotopes may have significant implications for the
very early history of the Solar System. 
FUN CAIs (and smaller grain hibonites) show roughly correlated anomalies
in $^{48}$Ca and $^{50}$Ti (see Fig. \ref{fig:correlation}, and
\cite{1978ApJ...220L..21L,1980ApJ...240L..73N}).
The anomalies are roughly correlated in the sense that CAIs or hibonites
that show excesses (relative to average Solar composition)
of $^{48}$Ca tend also to show excesses of $^{50}$Ti
while those that show deficits of $^{48}$Ca also show deficits of $^{50}$Ti.
A possible explanation for this is that the precursor dust of the Solar
System had a heterogeneous (but correlated) distribution of theses
isotopes.  As this precursor dust gathered to form the FUN CAIs, some
sampled an excess of the dust with large quantities of the neutron-rich
iron-group isotopes while others sampled a deficit of that dust.  The challenge
for Solar System studies is to characterize the precursor dust and mixing
in the early Solar nebula to understand the origin of the FUN CAIs.

%----------------------------------------------------------------------------
\section{Rare Ia Supernovae}\label{sect:Ia}
%----------------------------------------------------------------------------

The neutron-rich iron group isotopes are made in low entropy QSE (quasi
statistical equilibrium).  
In the QSE's, the isotopes like $\ca{48}$
are the dominantly produced ones. That the abundance of these isotopes
is low, then, tells us that their production must be rare.
And that environment happens during a thermonuclear supernovae explosion (Type
Ia).

%But also recall that massive stars can make ti50 and cr54 and note that
%there is evidence that anomalies in these isotopes came from injection
%into the early solar system (Larry Nittler's talk in Hawaii).

Speculate on the nature of the ejecta from such Ia events: either plating out
on pre-existing grains or condensing as metal droplets.

%----------------------------------------------------------------------------
\section{Galactic Chemical Evolution}\label{sect:gce}
%----------------------------------------------------------------------------

My thesis is that the rarity of the $^{48}$Ca-producing Ia's is reflected
in the FUN CAIs.  
Galactic Chemical Evolution (GCE) can simulate how dust grains
carry the neutron-rich isotopes and come to solar nebula.

These elements may all be related.  The rareness of the type Ia supernova
events may lead to a heterogeneous distribution of the neutron-rich iron-group
isotopes in the dust in the Galaxy.  Processing of that dust in the
interstellar medium will tend to homogenize the isotopes as supernova
shocks sputter atoms from the dust which then reaccrete onto other dust
grains.  If that homogenization process is not too efficient, however,
the precursor dust in the Solar System might indeed have been quite
isotopically heterogeneous so that the anomalies in the FUN CAIs could
be understood.

Unraveling the details of the history of the neutron-rich iron-group
isotopes is clearly a large problem that will require years, if not decades,
of work.  In this thesis I contribute to this grand problem by
focusing on the issue of production of these
isotopes in type Ia supernovae.  In particular, I developed some tools
to study the nucleosynthesis in low-entropy explosive environments in which
electron capture is changing the degree of neutron richness.

To carry out this project, we developed Nucnet Tools and Nucnet Projects.
I will describe the simple type Ia model and show the results in Chapter
\ref{chap:Ia}. This will give the initial abundances. Then I'll show how 
dusts grow calculations in the expanding materials of
the supernovae in Chapter \ref{chap:dust}. Here we will get the equilibrium
chemical form and the size of the grains that would condense.
Finally I will follow up with a simple GCE calculation to roughly estimate
the reservoirs in the Solar System and 
describe the future work in Chapter \ref{chap:conclusions}.
